 # Master the Art of FTOC: Comprehensive Examples and Expert Advice

Recent questions in FTOC grenivkah3z 2022-07-08

## What is the difference between first and second fundamental theorem of calculus? Sam Hardin 2022-07-05

## Given a continuous function $f:\left[0,1\right]\to \mathbb{R}$, prove that$\mathrm{\forall }t>0,\frac{1}{t}\cdot \mathrm{ln}\left({\int }_{0}^{1}{e}^{-tf\left(x\right)}dx\right)\le -minf\left(x\right).$ kolutastmr 2022-07-02

## Finding $f\left(x\right)$ in ${\mathrm{cos}}^{2}\left(x\right)f\left(x\right)={x}^{2}-2{\int }_{1}^{x}\mathrm{sin}\left(t\right)\mathrm{cos}\left(t\right)f\left(t\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}t$ Crystal Wheeler 2022-07-01

## How do you use the Fundamental Theorem of Calculus to find the derivative of $\int \frac{1}{1+{t}^{2}}dt$ from $x$ to $5$? pipantasi4 2022-07-01

## Finding the maximum of $G\left(x\right)={\int }_{x}^{x+a}f\left(t\right)\phantom{\rule{thinmathspace}{0ex}}dt$ using FTOC. lilmoore11p8 2022-07-01

## Find the derivative of the following function using the Fundamental Theorem of Calculus:$F\left(x\right)={\int }_{{x}^{3}}^{{x}^{6}}\left(2t-1{\right)}^{3}dt$ glitinosim3 2022-07-01

## Let $f$ be continuous on $I=\left[a,b\right]$ and let $H:I\to \mathbb{R}$ be defined by $H\left(x\right)={\int }_{x}^{b}f\left(t\right)dt,x\in I$. Find ${H}^{\prime }\left(x\right)$. Leland Morrow 2022-07-01

## Can you apply the fundamental theorem of calculus with the variable inside the integrand?For example: $\frac{d}{dx}\left({\int }_{a}^{x}xf\left(t\right)dt\right)$ Sarai Davenport 2022-06-28

## What is the difference between the two parts of FTOC? oleifere45 2022-06-25

## How does the first fundamental theorem of calculus guarantee the existence of antiderivatives of functions? Emanuel Keith 2022-06-25

## $f\left(x\right)={\int }_{0}^{\mathrm{sin}x}1+\mathrm{sin}\left(\mathrm{sin}\left(t\right)\right)dt$Find $\left({f}^{-1}{\right)}^{\prime }\left(0\right)$ excluderho 2022-06-24

## Derivative of a function and an integral$\frac{d}{dx}\left({x}^{6}\left({\int }_{0}^{sinx}\sqrt{t}dt\right)\right)$ Yahir Tucker 2022-06-24

## We know if $g$ is continuous on $\left(a,b\right)$ and $F\left(x\right)={\int }_{a}^{x}g\left(t\right)dt$, then${F}^{\prime }\left(x\right)=g\left(x\right)$But, how about if we have$F\left(x\right)={\int }_{a}^{h\left(x\right)}g\left(t\right)dt$What should ${F}^{\prime }\left(x\right)$ be?? can we still apply fundamental theorem of calculus? Petrovcic2x 2022-06-22

## Can use FToC to evaluate $\underset{x\to \mathrm{\infty }}{lim}\frac{{\int }_{0}^{x}\phantom{\rule{mediummathspace}{0ex}}f\left(t\right)dt}{{x}^{2}}$? Poftethef9t 2022-06-20

## If $f\left(x\right)$ is even, then what can we say about:${\int }_{-2}^{2}f\left(x\right)dx$If $f\left(x\right)$ is odd, then what can we say about${\int }_{-2}^{2}f\left(x\right)dx$Are they both zero? For the first one if its even wouldn't this be the same as${\int }_{a}^{a}f\left(x\right)dx=0$Now if its odd $f\left(-x\right)=-f\left(x\right)$. Would FTOC make this zero as well? watch5826c 2022-06-19

## Show that $g$ is differentiable and find ${g}^{\prime }\left(x\right)$, FTOC Carolyn Beck 2022-06-17

## Find the derivative of integral$H\left(x\right)={\int }_{3}^{{\int }_{1}^{x}{\mathrm{sin}}^{3}tdt}\frac{dt}{1+{t}^{2}+{\mathrm{sin}}^{6}t}$ juanberrio8a 2022-06-15

## Prove ${\int }_{-a}^{a}f\left(x\right)dx=0$assuming $f\left(x\right)$ is odd. Poftethef9t 2022-06-11
## Why does FTOC apply here, to find the derivative of ${\int }_{\mathrm{sin}\left(x\right)}^{\pi }\frac{t}{\mathrm{cos}\left(t\right)}dt$ anginih86 2022-06-08